What is the de Broglie wavelength of an oxygen molecule, O2 traveling at 521 m/s? Is the wavelength much smaller or much larger than the diameter of an atom. (on the order of 100 pm)

How do I do this

de Broglie wavelength = h/mv

h = Planck's constant in J*s.
m is mass in kg.
v is velocity in m/s

To calculate the de Broglie wavelength of a particle, you can use the following equation:

λ = h / p

Where λ is the de Broglie wavelength, h is the Planck's constant (6.62607015 × 10^-34 m^2 kg / s), and p is the momentum of the particle.

Before calculating the de Broglie wavelength, we need to determine the momentum of the oxygen molecule (O₂). The momentum (p) of an object can be calculated using the equation:

p = m * v

Where p is the momentum, m is the mass of the object, and v is the velocity.

The molar mass of oxygen, O₂, is approximately 32 g/mol. To convert this to kg/mol, we divide by 1000:

m = 32 g/mol / 1000 = 0.032 kg/mol

However, we are interested in the mass of a single oxygen molecule, so we divide this by Avogadro's number (6.02214076 × 10^23 mol^-1):

m = 0.032 kg/mol / (6.02214076 × 10^23 mol^-1) = 5.32 × 10^-26 kg

Now, we can calculate the momentum of the oxygen molecule:

p = m * v = (5.32 × 10^-26 kg) * (521 m/s) = 2.768 × 10^-23 kg·m/s

Finally, we can calculate the de Broglie wavelength:

λ = h / p = (6.62607015 × 10^-34 m^2 kg / s) / (2.768 × 10^-23 kg·m/s) = 2.392 × 10^-11 m = 23.92 pm

The calculated de Broglie wavelength is approximately 23.92 picometers (pm). Since this value is larger than the diameter of an atom (on the order of 100 pm), we can conclude that the wavelength is much larger than the atom's diameter.